Here we state and prove a decay result in the case of equal wave speeds propagation.
Define the state spaces
where
The associated energy term is given by
By a straightforward calculation, we have
From semigroup theory [25, 26], we have the following existence and regularity result; for the explicit proofs, we refer the reader to [16].
Lemma 2.1.
Let
be given. Then problem (1.1)–(1.5) has a unique global weak solution
verifying
We are now ready to state our main stability result.
Theorem 2.2.
Suppose that
and
. Then the energy
decays exponentially as time tends to infinity; that is, there exist two positive constants
and μ independent of the initial data and
, such that
The proof of our result will be established through several lemmas.
Let
where
is the solution of
Lemma 2.3.
Letting
,
,
,
,
be a solution of (1.1)–(1.5), then one has, for all
,
Proof.
By using the inequalities
and Young's inequality, the assertion of the lemma follows.
Let
Lemma 2.4.
Letting
,
,
,
,
be a solution of (1.1)–(1.5), then one has, for all
,
Proof.
Using (1.4) and (1.1), we get
The assertion of the lemma then follows, using Young's and Poincaré's inequalities.
Let
Lemma 2.5.
Letting
,
,
,
,
be a solution of (1.1)–(1.5), then one has, for all
,
Proof.
Using (1.3) and (1.5), we have
Then, using Young's and Poincaré's inequalities, we can obtain the assertion.
Next, we set
Lemma 2.6.
Letting
,
,
,
,
be a solution of (1.1)–(1.5), then one has, for all
,
Proof.
Letting
,
, then using (1.2), (1.3), we have
Noticing that
, then
Then, using Young's inequality, we can obtain the assertion.
We set
Lemma 2.7.
Letting
,
,
,
,
be a solution of (1.1)–(1.5), then one has, for all
,
Proof.
Let
,
, then using (1.1), (1.2), we have
Then, noticing
, again, from the above two equalities and Young's inequality, we can obtain the assertion.
Next, we set
Lemma 2.8.
Letting
,
,
,
,
be a solution of (1.1)–(1.5), then one has
Proof.
Using (1.1), (1.2), we have
Noticing (2.3) and (2.4), we have that
satisfy the following:
Similarly,
Then, insert (2.28) and (2.29) into (2.27), and the assertion of the lemma follows.
Now, we set
Lemma 2.9.
Letting
,
,
,
,
be a solution of (1.1)–(1.5), then one has, for all
,
Proof.
Using (1.5), we have
Then, using Young's and Poincaré's inequalities, we can obtain the assertion.
Now, letting
, we define the Lyapunov functional
as follows:
By using (2.4), (2.9), (2.13), (2.16), (2.19), (2.23), (2.26), and (2.31), we have
where
We can choose
big enough,
small enough, and
Then
are all negative constants; at this point, there exists a constant
, and (2.34) takes the form
We are now ready to prove Theorem 2.2.
Proof of Theorem 2.2.
Firstly, from the definition of
, we have
which, from (2.37) and (2.38), leads to
Integrating (2.39) over
and using (2.38) lead to (2.6). This completes the proof of Theorem 2.2.